One can then straightforwardly show basic properties of these unitaries (e.g. composition, commutation, and replacement of non-linear AND terms by linear XOR terms) using the rules of the ZH-calculus.
The phase polynomial formalism for \(m = 2\)  has been used extensively in studying optimisation problems for Clifford+T circuits [1]}, [2]}, [3]}, [4]}, and it was recently shown that all diagonal gates in the Clifford hierarchy are of the form \(U_\phi \) , where the level of the hierarchy depends on \(m\)  and the degree of \(\phi \)  [5]}. Gaining access to this phase polynomial structure diagrammatically could therefore yield new methods for quantum circuit optimisation and/or fault tolerant computation through automated diagram rewriting in tools like Quantomatic.
